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Example 1

Example 1.

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Example 1

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  1. Example 1 • Jim lives in New York City, but he plans to drive to Los Angeles in order to seek fame and fortune. Jim’s funds are limited, so he has decided to spend each night on his trip at a friend’s house. Jim has friends in Columbus, Nashville, Louisville, Kansas City, Omaha, Dallas, San Antonio, and Denver. Jim knows that after one day’s drive he can reach Columbus, Nashville, or Louisville. After two days of driving, he can reach Kansas City, Omaha, or Dallas. After three days of driving, he can reach San Antonio or Denver. Finally, after four days of driving, he can reach Los Angeles. In order to minimize the number of miles travelled, where should Jim spend each night of the trip?. The actual road mileages between cities are given in Figure below

  2. Example 1 - Network

  3. Example 1 - Solution

  4. Example 1 - Solution

  5. Example 1 - Solution

  6. Example 1 - Solution

  7. Computational Efficiency of DP • For the example described earlier, it would have been an easy matter to determine the shortest path from New York to Los Angeles by enumerating all possible paths • Since there are only 18 possible paths, use of DP did not really serve much purpose • For larger networks, DP is much more efficient than explicit enumeration of all possible solutions

  8. Example 2 – A Large Network

  9. Example 2: A large Network • There are 55 possible paths from node 1 to node 27 • Explicit enumeration of all paths require 5.55 = 15, 625 additions • With DP, determination of all ft(s) require 5x5=25 additions except f1(s) which requires 5 additions, where t is stage, and s is state • Thus DP requires only 25x4 + 5 = 125 additions • For larger networks, the computational savings achieved by DP are even more dramatic

  10. Example 3: A Reservoir Operation Problem A reservoir supplies water during a three month dry season during which there are no inflows. The net benefits to be derived from supplying various amounts of water during each of the three months is shown in the table below. The decision problem is to allocate the water available in storage at the beginning of the dry season over the three months, based on maximizing the total benefits. Determine the optimal policy assuming that three units of water is available in storage at the beginning of the dry season.

  11. Example 3: A Reservoir Operation Problem

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